Abstract

This paper is concerned with the reachability and controllability of fractional singular dynamical systems with control delay. The factors of such systems including the Caputo’s fractional derivative, control delay, and singular coefficient matrix are taken into account synchronously. The state structure of fractional singular dynamical systems with control delay is characterized by analysing the state response and reachable set. A set of sufficient and necessary conditions of controllability for such systems are established based on the algebraic approach. Moreover, an example is provided to illustrate the effectiveness and applicability of the proposed criteria.

1. Introduction

Singular systems play important roles in mathematical modeling of real-life problems arising in a wide range of applications. Depending on the area of application, these models are also called descriptor systems, semistate systems, differential-algebraic systems, or generalized state-space systems. In the past few decades, singular systems with integer-order derivative have been extensively studied due to the fact that singular systems describe practical systems better than normal ones (see [1–8] and references therein).

Fractional calculus and its applications to the sciences and engineering is a recent focus of interest to many researchers. Fractional differential equations have been proved to be an excellent tool in the modelling of many phenomena in various fields of engineering, physics, and economics. Actually, fractional differential equations are considered as alternative model to integer differential equations. Many practical systems can be represented more accurately through fractional derivative formulation. For more details on fractional calculus theory, one can see the monographs of Miller and Ross [9], Podlubny [10], Diethelm [11], and Kilbas et al. [12]. Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been paid more and more attention in [13–18] and references therein.

On the other hand, it is well-known that the issue of controllability plays an important role in engineering and control theory, which has close connections to pole assignment, structural decomposition, quadratic optimal control, and controller design, and so on. The problem of controllability for various kinds of dynamical systems with integer derivative has been extensively studied [1, 3–5, 19–22]. In particular, Dai [1] has established the theory of integer-order linear singular control systems. Yip and Sincovec [3] and Tang and Li [4] have investigated the controllability and observability of integer-order singular dynamical systems. It is worth mentioning that notable contributions have been made to fractional control systems [23–35]. The different techniques have been applied to investigate the controllability of various fractional dynamical systems, such as algebraic method [23, 24], geometrical analysis [25], fixed point theorems [26–30], and semigroup theory [31]. Recently, Sakthivel et al. [32–35] have investigated the approximate controllability problem for various kinds of fractional dynamical systems by using fixed point techniques, which have further enriched and developed the control theory of fractional dynamical systems. The earlier studies concerning the controllability of fractional dynamical systems with control delay can be found in [25, 27, 29], which especially focused on fractional normal systems. However, it should be emphasized that the control theory of fractional singular systems is not yet sufficiently elaborated, compared to that of fractional normal systems. In this regard, it is necessary and important to study the controllability problems for fractional singular dynamical systems. To the best of our knowledge, there are no relevant reports on reachability and controllability of fractional singular dynamical systems as treated in the current literature. Motivated by this consideration, in this paper, we investigate the reachability and controllability of fractional singular dynamical systems with control delay.

In this paper, we consider the reachability and controllability of the following fractional singular dynamical systems with control delay:
where denotes an order Caputo fractional derivative of , and ; , , , and are the known constant matrices, , , , and rank; is the state variable; is the control input; is the time control delay; and is the initial control function.

The main purpose of this paper is to establish reachability and controllability criteria for system (1) based on the algebraic approach. The factors of fractional singular dynamical systems including the Caputo’s fractional derivative, control delay, and singular coefficient matrices are taken into account synchronously. As discussed by Dai [1], we transform system (1) into two subsystems by the first equivalent form (FE1), which is more convenient to characterize the state reachable set. The state response, reachable set, and sufficient and necessary conditions for the controllability are obtained, respectively. The proposed criteria are applicable to a larger class of fractional dynamical systems. Therefore, we extend the known results [1, 23, 25] to a more general case.

This paper is organized as follows. In the next section, we recall some definitions and preliminary facts used in the paper. In Section 3, the state structure of system (1) is characterized by analysing the state response and reachable set. In Section 4, the necessary and sufficient conditions of controllability for two subsystems and system (1) are derived, respectively. In Section 5, an example is provided to illustrate the effectiveness and applicability of the proposed criteria. Finally, some concluding remarks are drawn in Section 6.

2. Preliminaries

In this section, we first recall some definitions of fractional calculus and preliminary facts. For more details, one can see [9–12]. Next, the first equivalent form (FE1) of system (1) is given.

(a) The Riemann-Liouville’s fractional integral of order with the lower limit zero for a function is defined as
provided that the right side is pointwise defined on , where is the Gamma function.

(b) The Caputo’s fractional derivative of order for a function is defined as
For ,

(c) The Mittag-Leffler function in two parameters is defined as
where , , and , denotes the complex plane. In particular, for ,
has the interesting property

(d) The Laplace transform of a function is defined as
where is -dimensional vector-valued function. For , it follows from [10] that

(e) For , the sequential fractional derivative for suitable function is defined by
where , and is any fractional differential operator; here we still mention it as .

Now, we introduce the first equivalent form (FE1) of system (1) by the nonsingular transform, which is also called the standard decomposition of a singular system.

Assume that is regular throughout this paper. From [1, 3], there exist two nonsingular matrices , such that system (1) is equivalent to canonical system as follows:
where , , , and is nilpotent whose nilpotent index is denoted by ; that is, , , and

Let , where denotes the integral part of . Let be the set of times piecewise continuously differentiable functions mapping the interval into . Moreover, denotes the set of times continuously differentiable functions mapping the interval into .

Applying the method in [3], we can obtain the precise form of the admissible initial state set for system (11). For , we have
Thus, the set of admissible initial data is

3. State Response and State Reachability

In this section, we characterize the state structure of system (11) by analysing the state response and reachable set.

Theorem 1. For any admissible initial data and the control function , the state response for system (11) can be represented in the following form:
where is the Mittag-Leffler function and is the sequential fractional derivative.

Proof. From [10], the Laplace transform of the order Caputo fractional derivative of function is
Taking the Laplace transform with respect to in both sides of the first equation of system (11), we obtain
Hence,
From [10], we have
Then (19) is equivalent to
The convolution theorem of the Laplace transform applied to (21) yields the form
Applying the inverse Laplace transform, we obtain
On the other hand, inserting (16) into both sides of the second equation in system (11) yields
Then (16) is a solution of the second equation in system (11).Next, we will show that any solution of the second equation in system (11) has the form of (16). Define linear operator by . Then the second equation in system (11) becomes
Since is nilpotent, and can exchange, and , then we have
Thus, (16) is indeed a solution of the second equation in system (11).Therefore, the proof of Theorem 1 is completed.

For convenience sake, some notations are introduced as follows.

Given matrices and , denote as the range of ; that is,
Let
Then the space is spanned by the columns of
and the space is spanned by the columns of
For any nonzero polynomial , we define by
where denotes the matrix transpose.

The following lemma is required for the main result, which is a natural extension of Lemma 2-1.1 in [1].

Lemma 2. Given matrices and , then

Proof. The formula (32) holds if and only if
If and , , that is,
then we have
Since polynomial only has a finite number of zero on and , we immediately have
Repeatedly taking the Caputo’s derivative on both sides of (37), we have
Substituting in (39) yields
which implies that , . According to Cayley-Hamilton theorem [4], can be represented as
For , it follows from (39) and (41) that
Inserting (42) into (38) yields
Repeatedly taking the Caputo’s fractional derivative on both sides of (43) and letting , we have
which implies that , . Therefore, we have
Conversely, if and , then , , , and
For , it follows from (41) that
For , the same argument yields
Hence, , which implies
From (45) and (49), we know that (32) is true. The proof is therefore completed.

Definition 3. Any vector in -dimensional vector space is said to reachable if there exists an admissible initial data , admissible control input , and such that the solution of system (1) (or (11)) satisfies .Let be the reachable set from any admissible initial data . Then

Considering the reachable set from the initial conditions and , we have the following theorem.

Theorem 4. For system (11), the reachable set can be represented as
where “” is the direct sum in vector space.

Proof. Let , from (15), (16), and (50); then there exists and such that
Combining (41) and (52) yields
Therefore, . Moreover, we also have
which implies that . Thus,
On the other hand, we need to prove
Suppose that
where , , and , . For the initial conditions and , let ; then we have
As discussed by Yip and Sincovec [3], we choose to satisfy
Then contributes nothing to at and . Now, we prove that (60) is true.In fact, for , from Lemma 2, there exists such that .Let
Then
Therefore, (60) is true.For , there exist such that
There exists a function such that , , and . Let
then we have
and . It follows from (60) and (65) that . Thus, (56) holds. Combining (55) and (56), we know that (51) is true. Then the proof of Theorem 4 is completed.

4. Controllability Criteria

In this section, we proceed to investigate the controllability criteria of system (1) and system (11) based on the reachable set. A set of sufficient conditions and necessary conditions for the controllability are derived based on the algebraic approach.

Definition 5. System (1) (or (11)) is said to be controllable at if one can reach any state at from any admissible initial data .

Proof. We firstly prove the necessity of Theorem 6. If system (11) is controllable, for any , to the initial state and initial control function , there exists and a control function such that could be written in the form of (58) and (59). That is
which implies that
Obviously,
Consequently, we have
Thus, (66) and (67) are true.Next, we prove the sufficiency of Theorem 6. If (66) and (67) are true, then we know that (71) holds. For any and any initial state and initial control function , let
For , we have . Then there exists a control such that
Let , then we have
Thus, by Definition 5, system (11) is controllable. Therefore, the proof is completed.

Applying the results of Yip and Sincovec [3], Tang and Li [4], and Theorem 6 to the pair of matrices, we can obtain the following results.

Remark 9. If we choose and , then system (1) reduces to an integer-order singular system without control delay. From Theorem 8, system (1) is controllable if and only if
which is just the result in [1].

If we choose , , then system (1) reduces to a fractional normal system without control delay. From Theorem 8, system (1) is controllable if and only if
which is just Corollary 3.4 in [23].

If we choose , then system (1) reduces to a fractional normal system with control delay. From Theorem 8, system (1) is controllable if and only if
which is just Theorem 3 in [25]. Therefore, our results extend the existing results [1, 23, 25].

Remark 10. Theorem 8 actually offers an algebraic criterion of the exact controllability for linear fractional singular dynamical systems, which is concise and convenient to check the controllability of such systems. Motivated and inspired by the works of Sakthivel et al. [32–35], the approximate controllability of linear (nonlinear) fractional singular dynamical systems with the control delay will become our future investigative work.

5. An Illustrative Example

In this section, we give an example to illustrate the applications of Theorem 8.

Example 11. Consider the controllability of fractional singular dynamical systems as follows:
where the admissible initial data . Now, we apply Theorem 8 to prove that system (81) is controllable. Let us take
If we choose such that , then is regular. By using the elementary transformation of matrix, one can obtain
Thus, by Theorem 8, system (81) is controllable.

6. Conclusions

In this paper, the reachability and controllability of fractional singular dynamical systems with control delay have been investigated. The state structure of fractional singular dynamical systems with control delay has been characterized by analysing the state response and reachable set. A set of sufficient and necessary conditions of controllability criteria for such systems has been established based on the algebraic approach. An example is also presented to illustrate the effectiveness and applicability of the results obtained. Both the proposed criteria and the example show that the controllability property of fractional linear singular dynamical systems is dependent neither on the order of fractional derivative nor on control delay. Comparing with some existing results [1, 23, 25], we find that the algebraic approach has been extended to consider the controllability of more general fractional singular dynamical systems. Motivated and inspired by the works of Sakthivel et al. [32–35], the approximate controllability of linear (nonlinear) fractional singular dynamical systems will become our future investigative work.

Conflicts of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the Associate Editor, Professor Han H. Choi, and the anonymous reviewers for their helpful and valuable comments and suggestions which improved the quality of the paper. This work is jointly supported by the National Natural Science Foundation of China under Grant nos. 11071001, 11072059, and 61272530, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Programs of Educational Commission of Anhui Province of China under Grant nos. KJ2011A197 and KJ2013Z186. This work is also funded by the Deanship of Scientific Research, King Abdulaziz University (KAU), under Grant no. 3-130/1434/HiCi. Therefore, the authors acknowledge technical and financial support of KAU.